Problem 44 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

CONSUMERS' SURPLUS In Exercises 43 through 46, \(p=D(q)\) is the demand curve for a particular commod- ity; that is, \(q\) units of the commodity will be demanded when the price is \(p=D(q)\) dollars per unit. In each case, for the given level of production \(q_{0}\), find \(p_{0}=D\left(q_{0}\right)\) and compute the corresponding consumers' surplus. . \(D(q)=100-4 q-3 q^{2} ; q_{0}=5\) units

Step-by-Step Solution

Answer

The consumers' surplus is 300.

Step 1 - Find the Price at Given Level of Production

To find the price at the given production level, substitute 5 for the quantity \( q \) in the demand function. The demand function is given as: \[ D(q) = 100 - 4q - 3q^2 \] Substitute \( q_0 = 5 \): \[ p_0 = D(5) = 100 - 4(5) - 3(5)^2 \] Calculate \( p_0 \): \[ p_0 = 100 - 20 - 75 = 5 \] So, the price \( p_0 \) at the production level of 5 units is 5 dollars.

Step 2 - Set up the Consumers' Surplus Formula

The consumers' surplus is found using the formula: \[ CS = \int_{0}^{q_0} D(q) \, dq - p_0 q_0 \] where \( q_0 = 5 \) and \( p_0 = 5 \).

Step 3 - Compute the Definite Integral

To compute the integral, set up the integral of the demand function from 0 to 5: \[ \int_{0}^{5} (100 - 4q - 3q^2) \, dq \] Evaluate the integral: 1. Integrate each term separately: \[ \int 100 \, dq = 100q \] \[ \int -4q \, dq = -2q^2 \] \[ \int -3q^2 \, dq = -q^3 \] Combine and evaluate at the bounds 0 and 5: \[ \left[ 100q - 2q^2 - q^3 \right]_0^5 = (100(5) - 2(5)^2 - (5)^3) - (100(0) - 2(0)^2 - (0)^3) \] \[ = (500 - 50 - 125) - (0) \] \[ = 325 \]

Step 4 - Apply the Consumers' Surplus Formula

Apply the computed integral value and the values of \( p_0 \) and \( q_0 \) in the consumers' surplus formula: \[ CS = 325 - (5 \times 5) \] Calculate the final value: \[ CS = 325 - 25 = 300 \]

Key Concepts

demand curve

The demand curve represents the relationship between the price of a commodity and the quantity demanded by consumers. In mathematical terms, it is usually expressed as a function of price in terms of quantity demanded, such as:
\( D(q) = 100 - 4q - 3q^2 \).
In this example, the demand curve is a quadratic equation showing how the demand decreases as the price increases.
Understanding the demand curve helps us predict how changing prices could affect the quantity demanded. This is essential in economic analysis and in setting prices for products.

definite integral

The definite integral is a fundamental concept in calculus that calculates the net area under a curve over a particular interval. In our context, it is used to find the total value of the demand over a range of quantities.
\[ \text{The integral we computed:} \ \int_{0}^{5} (100 - 4q - 3q^2) \ dq \]
This integral evaluates the sum of the areas under the demand curve from 0 to 5 units of the commodity. Integrals help in summing up continuously varying quantities, such as the total consumer willingness to pay across different quantities.
By solving this integral, we translate a complex curve into a simpler numerical value representing the aggregate economic activity over that range.

price calculation

Calculating the price for a given quantity involves substituting that quantity into the demand function. For example, if we want to find the price at a production level of 5 units:
\[ p_0 = D(5) = 100 - 4(5) - 3(5)^2 \]
This calculation gives us the price at which 5 units will be demanded. Understanding this concept allows businesses to set prices based on how much consumers are willing to pay at different production levels.
More generally, it helps in setting pricing strategies, predicting revenues, and understanding market behavior. Accurate price calculation is essential for effective financial planning and analysis.

economic surplus

Economic surplus is a measure of the benefit that consumers and producers receive from participating in the marketplace. It consists of consumer surplus and producer surplus. In this exercise, we are focused on consumer surplus, which is the difference between what consumers are willing to pay and what they actually pay.

• **Formula for Consumer Surplus (CS):� \[ CS = \int_{0}^{q_0} D(q) dx - p_0 q_0 \]
  By computing the definite integral of the demand function and subtracting the total expenditure (price multiplied by quantity), we determine the consumer surplus. It represents the aggregate extra benefit consumers get because they pay less than what they are willing to.
  In our example, the consumer surplus came out to be 300, reflecting the overall gain to consumers in that market. Understanding economic surplus helps in policy-making and market analysis, providing insights into market efficiency and consumer welfare.